public 43:08

James Colliander : Crowdmark presentation

  -   Presentations ( 327 Views )

public 01:34:43

Bruce Donald : Some mathematical and computational challenges arising in structural molecular biology

  -   Applied Math and Analysis ( 304 Views )

Computational protein design is a transformative field with exciting prospects for advancing both basic science and translational medical research. New algorithms blend discrete and continuous mathematics to address the challenges of creating designer proteins. I will discuss recent progress in this area and some interesting open problems. I will motivate this talk by discussing how, by using continuous geometric representations within a discrete optimization framework, broadly-neutralizing anti-HIV-1 antibodies were computationally designed that are now being tested in humans - the designed antibodies are currently in eight clinical trials (See https://clinicaltrials.gov/ct2/results?cond=&term=VRC07&cntry=&state=&city=&dist= ), one of which is Phase 2a (NCT03721510). These continuous representations model the flexibility and dynamics of biological macromolecules, which are an important structural determinant of function. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. These distributions are not fully constrained by the limited information from experiments, making the problem ill-posed in the sense of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem must be regularized by making (hopefully reasonable) assumptions. I will present new ways to both represent and visualize correlated inter-domain protein motions (See Figure). We use Bingham distributions, based on a quaternion fit to circular moments of a physics-based quadratic form. To find the optimal solution for the distribution, we designed an efficient, provable branch-and-bound algorithm that exploits the structure of analytical solutions to the trigonometric moment problem. Hence, continuous conformational PDFs can be determined directly from NMR measurements. The representation works especially well for multi-domain systems with broad conformational distributions. Ultimately, this method has parallels to other branches of applied mathematics that balance discrete and continuous representations, including physical geometric algorithms, robotics, computer vision, and robust optimization. I will advocate for using continuous distributions for protein modeling, and describe future work and open problems.

public 01:14:48

Robert V. Kohn : A Variational Perspective on Wrinkling Patterns in Thin Elastic Sheets: What sets the local length scale of tensile wrinkling?

  -   Gergen Lectures ( 282 Views )

The wrinkling of thin elastic sheets is very familiar: our skin wrinkles, drapes have coarsening folds, and a sheet stretched over a round surface must wrinkle or fold.

What kind of mathematics is relevant? The stable configurations of a sheet are local minima of a variational problem with a rather special structure, involving a nonconvex membrane term (which favors isometry) and a higher-order bending term (which penalizes curvature). The bending term is a singular perturbation; its small coefficient is the sheet thickness squared. The patterns seen in thin sheets arise from energy minimization -- but not in the same way that minimal surfaces arise from area minimization. Rather, the analysis of wrinkling is an example of "energy-driven pattern formation," in which our goal is to understand the asymptotic character of the minimizers in a suitable limit (as the nondimensionalized sheet thickness tends to zero).

What kind of understanding is feasible? It has been fruitful to focus on how the minimum energy scales with sheet thickness, i.e. the "energy scaling law." This approach entails proving upper bounds and lower bounds that scale the same way. The upper bounds tend to be easier, since nature gives us a hint. The lower bounds are more subtle, since they must be ansatz-free; in many cases, the arguments used to prove the lower bounds help explain "why" we see particular patterns. A related but more ambitious goal is to identify the prefactor as well as the scaling law; Ian Tobasco's striking recent work on geometry-driven wrinkling has this character.

Lecture 1 will provide an overview of this topic (assuming no background in elasticity, thin sheets, or the calculus of variations). Lecture 2 will discuss some examples of tensile wrinkling, where identification of the energy scaling law is intimately linked to understanding the local length scale of the wrinkles. Lecture 3 will discuss our emerging undertanding of geometry-driven wrinkling, where (as Tobasco has shown) it is the prefactor not the scaling law that explains the patterns seen experimentally.

public 01:49:44

DOmath 2022 students : DOmath final presentations II

  -   Undergraduate Seminars ( 272 Views )